Download P-values

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Chapter 8.4: P-value
Instructor: Dr. Arnab Maity
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Example: Recall the aspirin example in the previous chapter. Let p denote the proportion
of all aspirin-taking men in this age group who would have a heart attack over the next
3 years. To test H0 : p = 0.17 versus Ha : p < 0.17, we obtain the test statistic value
z = −1.597.
• The following table displays the zα value for different significance levels α. Please fill
in the blanks.
Sig. Level α
0.01
0.05
0.06
0.10
zα Rejection region
2.33
z < −2.33
1.65
z < −1.65
1.56
1.28
z < −1.28
Conclusion
Do not reject H0
Do not reject H0
Do not reject H0
• Suppose Z follows N (0, 1). Calculate P (Z < −1.597).
• What is the smallest level at which H0 would be rejected?
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Definition: The P-value is the smallest level of significance at which H0 would be
rejected.
Conclusions based on the p-value:
1. If p-value ≤ α then Reject H0 at level α.
2. If p-value > α then Do not reject H0 at level α.
Another Interpretation: The P-value is the probability of obtaining a test
statistic value as extreme as the observed value, calculated assuming H0 is true.
Visual interpretation for Z tests.
Some facts about P-values
• P-value is a probability
• This probability is calculated assuming that the null hypothesis H0 is true.
• Caution: The P-value is NOT the probability that H0 is true, nor is it an error
probability!
• To determine the P-value, we must first decide which values of the test statistic are as
extreme as the value obtained from our sample.
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P -values for z-tests: Here the test statistic Z follows a N (0, 1) distribution.
Alternative Hypothesis Rejection Region for Level α Test
Two-tailed test
P-value = 2(1 − Φ(|z|))
Upper-tailed test
P-value = 1 − Φ(z)
P-value = Φ(z)
Lower-tailed test
where Φ(z) is the CDF of a standard normal distribution.
P -values for t-tests: Here the test statistic T follows a tn−1 distribution. The
observed value of the test statistic is t.
Alternative Hypothesis Rejection Region for Level α Test
Two-tailed test
P-value = 2(1 − P r(T < |t|))
Upper-tailed test
P-value = 1 − P r(T < t)
P-value = P r(T < t)
Lower-tailed test
1. (Recall the gas mileage example) A car manufacturer claims that, when driven at a
speed of 50 miles per hour, the mileage of a certain model follows a normal distribution
with mean 30 miles per gallon. A random sample of 10 cars yields x = 29.4 miles per
gallon, with sample standard deviation s = 4 miles per gallon. Calculate the p-value.
Is there a reason to believe that the manufacturer is overestimating average mileage
(with significance level of 0.01)?